If you use the 'Forwarders' menu of your cPanel to forward email from aaa@bbb.com to xxx@yyy.com, all email sent to aaa@bbb.com will not just be forwarded to xxx@yyy.com, but will also be saved in the mailbox of aaa@bbb.com (unless you haven't created an email account for aaa@bbb.com in the 'Email Accounts' section of your cPanel).

This could be a problem if, for example, you have little free disk space available on your account.

To prevent email forwarded to xxx@yyy.com from being saved in the mailbox of aaa@bbb.com, do not use the Forwarders menu. Use the User Level Filtering > Manage Filters > Create a new Filter menu of your cPanel instead. Create a filter to forward and then discard email, like so:

The above will affect all mail sent to aaa@bbb.com. If you only want to forward a specific subset of mail sent to aaa@bbb.com, change the 'Rules' accordingly.

Note: while simply deleting your aaa@bbb.com email account can be a solution to this problem in some cases, in other cases it is preferable to keep an email account for aaa@bbb.com. An example of the latter: when you want to reply to mail forwarded from aaa@bbb.com to xxx@yyy.com with your aaa@bbb.com address, it is a good idea to send it through the SMTP server of aaa@bbb.com (in order for the reply to not appear as a redirect to the recipient, as would happen in Outlook clients etc.). To use the SMTP server of aaa@bbb.com, you need to have an email account for it. This option is also available for Gmail users.